Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein (eBook)

Front Cover 1
Foundation of Euclidean and Non-Euclidean Geometries According to F. Klein 4
Copyright Page 5
Table of Contents 6
PREFACE 10
CHAPTER 1. AXIOMS 12
§ 1. Axioms of incidence 12
§ 2. Axioms of betweenness 13
§ 3. Axiom of continuity 14
§ 4. Axioms of motion 14
CHAPTER 2. CONSEQUENCES OF THE SYSTEM OF AXIOMS I 16
§ 5. Simple properties of straight lines and planes 16
§ 6. Desargues configurations 19
§ 7. Linear subspaces 22
§ 8. The lattice of linear subspaces 23
§ 9. Basic projective configurations 24
§ 10. Projection and intersection 26
CHAPTER 3. SIMPLE CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II 28
§ 11. Segments. Triangles 28
§ 12. Properties of segments 31
§ 13. Linear ordering 34
§ 14. Properties of triangles 40
§ 15. The tetrahedron 44
§ 16. Neighbourhoods 47
§ 17. Validity of the systems of Axioms I, II for the basic domain R' 48
§ 18. Generalization of the notion of space 50
§ 19. The extension and restriction of spaces 50
CHAPTER 4. PROJECTIVE CLOSURE 51
§ 20. Half-subspaces 51
§ 21. Half-pencils. Angles 56
§ 22. Some properties of pencils and bundles 61
§ 23. Coplanar Desargues configurations 62
§ 24. Improper pencils of lines 66
§ 25. Improper bundles of lines 72
§ 26. The projective closure R of R 76
§ 27. The projective axioms 95
§ 28. The general case 104
CHAPTER 5. INVESTIGATION OF THE PROJECTIVE SPACE 107
§ 29. Preliminaries 107
§ 30. Theorem of duality in projective space 109
§ 31. Collineations 110
§ 32. The Erlangen programme 113
§ 33. Theorem of duality of the plane 114
§ 34. Perspectivities and projectivities 116
§ 35. Central collineations of the plane 121
§ 36. Separation 133
§ 37. Cyclic ordering 144
§ 38. Projective segments and angles 149
§ 39. Complete quadrangles. Harmonic points 155
§ 40. Preliminaries about coordinate systems 164
§ 41. Coordinates in projective scales 174
§ 42. Halving a projective scale 179
§ 43. Coordinates for dyadic sets of points on a line 181
CHAPTER 6. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III 186
§ 44. Preliminaries 186
§ 45. Theorem concerning the infinite point 191
§ 46. Coordinates in an affine line 196
§ 47. Coordinates on the basic projective configurations of the first degree 202
§ 48. Point-coordinates in an affine plane 205
§ 49. The fundamental theorem of projective geometry 214
§ 50. Point-coordinates in an affine space 223
§ 51. Vectors 229
§ 52. Homogeneous point- and plane-coordinates in space. Point- and line-coordinates in a plane 231
§ 53. Determination of all collineations of the space 241
§ 54. Determination of the coordinate transformations of space 245
§ 55. Transformation of projective coordinates 250
§ 56. Cross ratio 252
§ 57. Imaginary points 258
§ 58. Fixed elements of projectivities 259
§ 59. Involutions 260
§ 60. Involutory collineations of a plane 263
CHAPTER 7. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III, IV 265
§ 61. Extended motions 265
§ 62. The comparability of segments 270
§ 63. Reflections and rotations. Absolute polar plane 275
§ 64. Metric scales. Infinite and ultra-infinite points. Elliptic, parabolic and hyperbolic geometries 286
§ 65. Absolute involution of points on a proper line 294
§ 66. Midpoint and bisector 299
§ 67. The lines perpendicular to a proper plane 304
§ 68. Motions as products of reflections 312
§ 69. Polarities with respect to surfaces and curves of the second order 314
§ 70. The absolute configuration in the elliptic case 322
§ 71. The absolute configuration in the hyperbolic case 324
§ 72. Characterization of motions in the non-parabolic case 333
§ 73. The absolute configuration and characterization of motions in the parabolic case 336
§ 74. Formulae of motion of the three geometries 348
§ 75. The consistency of the three geometries 363
§ 76. Measuring of segments 377
§ 77. Measuring of angles 387
§ 78. Applications to trigonometry 393
Bibliography 402
Index 404